Question: Khaled and Wilma were asked to find an explicit formula for the sequence $1\,,\,3\,,\,9\,,\,27,...$, where the first term should be $f(1)$. Khaled said the formula is $f(n)=1\cdot3^{{n-1}}$, and Wilma said the formula is $f(n)=1\cdot3^{{n}}$. Which one of them is right? Choose 1 answer: Choose 1 answer: (Choice A) A Only Khaled (Choice B) B Only Wilma (Choice C) C Both Khaled and Wilma (Choice D) D Neither Khaled nor Wilma
In a geometric sequence, the ratio between successive terms is constant. This means that we can move from any term to the next one by multiplying by a constant value. Let's calculate this ratio over the first few terms: $\dfrac{27}{9}=\dfrac{9}{3}=\dfrac{3}{1}={3}$ We see that the constant ratio between successive terms is ${3}$. In other words, we can find any term by starting with the first term and multiplying by ${3}$ repeatedly until we get to the desired term. Let's look at the first few terms expressed as products: $n$ $1$ $2$ $3$ $4$ $f(n)$ ${1}\cdot\!{3}^{0}$ ${1}\cdot\!{3}^{1}$ ${1}\cdot\!{3}^{2}$ ${1}\cdot\!{3}^{3}$ We can see that every term is the product of the first term, ${1}$, and a power of the constant ratio, ${3}$. Note that this power is always one less than the term number $n$. This is because the first term is the product of itself and plainly $1$, which is like taking the constant ratio to the zeroth power. Thus, we arrive at the following explicit formula (Note that ${1}$ is the first term and ${3}$ is the constant ratio): $f(n)={1}\cdot{3}^{{\,n-1}}$ So Khaled is definitely right. What about WIlma? We can see that $f(n)=1\cdot3^{{\,n}}$ is not a correct formula, because the constant ratio is multiplied one extra time for each term. For instance, according to this formula, the value of the first term would be: $f(1)=1\cdot3^{{\,1}} = 3$. However, according to our table of values, $f(1)=1$. So WIlma is definitely wrong. Only Khaled got a correct explicit formula.